Local models for conical Kähler-Einstein metrics

Abstract

In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.

Bubbling of Kähler-Einstein metrics

“…A version of this conjecture was formulated in a discussion with Cristiano Spotti in 2016. For low dimensional examples, progress has been made by him and Martin de Borbon [13].…”

Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

“…on the region {0 < h < 1}. See [7,8] for two recent treatments of this metric in the literature. Fixing an arbitrary point p ∈ D ⊂ L, we choose local holomorphic coordinates (z 1 , .…”

“…We begin by explaining an alternative description of complex hyperbolic cusps in terms of the Calabi ansatz [7,8]. Let D be a (necessarily projective) complex torus of complex dimension n − 1 together with a negative holomorphic line bundle L → D. Let h be any Hermitian metric on L such that the curvature form of the Hermitian metric dual to h is a flat Kähler metric on D. Then h is unique up to scaling, i.e., up to replacing h by λh for some λ ∈ R + .…”